Approximate Conservation Laws of Nonvariational Differential Equations

Abstract

The concept of an approximate multiplier (integrating factor) is introduced. Such multipliers are shown to give rise to approximate local conservation laws for differential equations that admit a small perturbation. We develop an explicit, algorithmic and efficient method to construct both the approximate multipliers and their corresponding approximate fluxes. Our method is applicable to equations with any number of independent and dependent variables, linear or nonlinear, is adaptable to deal with any order of perturbation and does not require the existence of a variational principle. Several important perturbed equations are presented to exemplify the method, such as the approximate KdV equation. Finally, a second treatment of approximate multipliers is discussed.